Breaking beliefs: Can you “believe” and “not believe” at the same time?

One thing I like to do is try to break shit. I’m actually kinda good at it. As a kid I was very good at taking things apart, but not so good at putting them back together. Now, with a little more experience and knowledge under my belt I still like to break things, but now I can also put them back together as well. So what about trying to break things about what we believe and putting them back together in some type of cogent way?

Recently Matt Dillahunty from the Atheist Experience and I had a little Twitter back and forth on beliefs, and one of the things he said was that basically “You believe x or you do not believe x”…and in classical logic that is completely correct:

Bap V ~Bap

Which means an agent “a” either believes (B) that the proposition is true or “V” does not “~” believe the proposition is true. But let’s take an example of contradictory beliefs:

Bap V Ba~p

Here we have a situation where the agent either beliefs p is false or believes p is true, but this is not an actual dichotomy since clearly the agent can believe neither that p is true nor false…or can hold conflicting beliefs that both are true. While this is a state of irrationality, it could be that the agent just is not aware he is holding two conflicting beliefs or he could just accept that he is in an irrational state, but in any case, it is not logically impossible. Remember, there is a distinct difference between the ontology here (being, or state of affairs of the universe) and the epistemology (How or what we know about something) of the situation.

We could agree ontologically that given some (not all) propositions must either be T or F as in p=”The Earth is flat”. Assuming flat means having general flatness and not “spherical” as science shows it is, then that statement is ontologically true or false such that it is or is not the case that the state of affairs of the universe are such that the Earth is flat. What one believes about that however can vary, the agent can believe p is true, that p is false, that p isn’t even a proposition (noncognitivist), or the agent may not believe p is true nor believe p is false, or the agent may not be aware of what his position is on the issue. It is possible to have beliefs that we are not aware of, or merely not actively recognize as being contradictory beliefs (Armstrong (1969)) in which case the agent just may not know what he believes, or does not believe, even if ontologically he either does or does not believe p.

Now this is where I see it to start to get interesting to see if we can break beliefs here…

If the agent holds two competing beliefs such that they are hold both Bap and Ba~p then:

(i) It generally is the case that an agent who believes p is false implies the agent does not believe p is true (Ba~p →~Bap) and an agent who believes p is true implies the agent does not believe p is false. (Bap → ~Ba~p)

(ii) It also is generally, in a loose case, that linguistically when an agent says they do not believe p is true, it may sometimes be inferred to mean the agent believes p is false…or that they have no belief either way. This would be then, by weak inference, then that ~Bap →Ba~p, but clearly not in any strong case nor by entailment nor necessity. (In a more strict interpretation of an agent saying they do not believe p is true then ~Bap ↛ Ba~p or if an agent does not believe p is false then ~Ba~p ↛ Bap). (↛ to mean does not imply)

So what happens if a person does have conflicting beliefs such that they hold both p to be true and “^” p to be false (Bap ^ Ba~p). We have already established that this is not a logical impossibility, but merely an irrational state. If the agent holds to Bap then from (i) and (ii) they also hold to ~Bap (agent believes p is true then agent does not believe p is false). If the agent holds to Ba~p then from (i) and (ii) they also hold to ~Ba~p (agent believes p is false then agent does not believe p is true). Now we have a state where the agent seemingly holds to (Bap ^ ~Ba~p) ^ (~Bap & ~Bap) or:

“Agent believes p is both true and false, and does not believe p is true and does not believe p is false)”

We can then reduce that to say “Agent believes p is true and does not believe p is true”.

At 3 AM you decide if I just broke belief or not…

(There is probably most likely something fundamentally wrong here, but at 3 AM it is just fun to muse about these things…so please feel free to rip it apart as you will)